Caution: remove the batteries from your test setup when it is not in use. Two of the batteries shown in your test setup are likely to leak corrosive chemicals into the battery holder if allowed to discharge completely and remain for a long time.
You claim that the current does not increase with the parallel battery. How certain are you of this? What if the current increase is very slight, and is less than the ability of your meter to measure?
Here is your assignment:
-- get a more accurate meter, the best you can borrow
-- measure the current with battery #1 alone.
-- measure the current with battery #2 alone.
-- measure the current with both batteries in parallel.
I predict that the third measurement will be greater than the minimum of the first two.
It is common in indroductory electronics to consider batteries as a source of voltage. But batteries are really very complex devices for which the voltage source is only an approximation. A better approximation is a voltage source with a low-value resistor in series. You can actually estimate the value of this series resistor by measuring the voltage drop when a load is applied to a battery and using the usual rules for series resistors and ohm's law. Most electrical engineering stops with this, but there are even more complex models for a battery that represent its behavior even more precisely.
Learn all you can, but always be skeptical and ask challenging questions while I recharge my cell phone.